Question: Determine how many solutions exist for the system of equations. ${-9x-3y = 15}$ ${y = 7-2x}$
Solution: Convert both equations to slope-intercept form: ${-9x-3y = 15}$ $-9x{+9x} - 3y = 15{+9x}$ $-3y = 15+9x$ $y = -5-3x$ ${y = -3x-5}$ ${y = 7-2x}$ ${y = -2x+7}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -3x-5}$ ${y = -2x+7}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.